Helix formation by d(TA) oligomers

J. Mol. Biol. (1970) 48, 145-171

Helix Formation

by d(TA) Oligomers

II.7 Analysis of the Helix-Coil Transitions and Circular Oligomers

of Linear

IMMO E. SCHEFFLER~, ELLIOT L. ELSON~~AND ROBERT L. BALDWIN

Department of Biochemistry, Stanford University School of Me&&e Stanford, Calif. 94305, U.S.A. and Department

of Chemidry, University of Cdifomia

San Diego, La Jolla, Calij. 92038, U.S.A. (Received 18 July 1969, and in revised form

7 October 1969)

An experimental study has been made of the factors determining the co-operativity of DNA melting, using two types of d(TA) oligomers (linear and circular) with the alternating base sequence. . . ATAT. . . . The linear ohgomers form open hairpin helices which melt chiefly from the open end. The circular oligomers form closed hairpin h&es with a loop at each end and melt by enlarging the loops. Analysis of the melting curves for open hairpins (in O-6 x-N&+, to avoid chainlength dependent electrostatic effects) yields y, the equilibrium constant for closing the minimum-size loop: y = 0,003. Interpretation of the equilibrium constant for loop closure is discussed; it can be expressed as the equilibrium constant for formation of an isolated base pair in a bimolecular reaction multiplied by the effective concentration of one base in the vicinity of the other, before the loop is closed. The effective concentration is related to loop size by the loopweighting function. The melting curves of closed hairpin helices can be used to test the formulation of the loop-weighting function. The results show that for small DNA loops the loop-weighting function diiers from that predicted by Jacobson & Stockmayer (1950) for long gaussian chains: the differences are of the type expected for short chains with hindered rotation. Although the closed hairpin helices contain two loops and the open hairpins have only one, nevertheless the closed hairpins are more stable. The reason is a large difference in the conformational entropies of the two random-chain forms: the open circle can assume only a fraction of the conformations av&lable to the linear chain. Since the midpoint of the melting curve (the T,) depends on the relative stability of helix to random chain, the helix formed by the circular oligomer has the higher T,. The T, values of the closed hairpins are higher than predicted by earlier theories: higher, in fact, than the T, of poly d(AT). The reason is that the minimum-size loops in a closed hairpin helix form more readily thsn use of the Jacobson-Stocknmyer loop-weighting function would predict, t Yapur I in this series is Scheffler, Elson & Baldwin, 1968. $ Present address: Department of Biological Chemistry, Harvard Medical School, Boston, Mass. 02113, U.S.A. \I Present adclress: Department of Chemistry, Cornell University, Ithaca, N.Y. 14850, U.S.A. 146 10

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1. Introduction The statistical theory of DNA “melting” (i.e., the transition from helix to random polymer chains) has been well understood for a number of years (cf. Zimm, 1960). However, the parameters which determine the co-operativity of melting have been difficult to measure, even when the problem is simplified by studying synthetic DNA’s with only one type of base pair, because most measurable quantities depend on more than one parameter. Also it has been difficult to test the basic assumptions of the theory. A crucial assumption is the use of the Jacobson-Stockmayer (1950) loopweighting function to express the equilibrium constant for loop closure; this function is known to be valid only for long polymer chains whose conformations follow a gaussian distribution. It is possible to study loop formation in a simple model system by using oligonucleotides which have a repeating and self-complementary base sequence: such oligomers can form a DNA helix with a loop at one end (“hairpin helices”) from a single oligonucleotide strand. In the preceding paper we defined conditions in which d(TA), oligomers (5 < N < 25) form hairpin helices (Scheffler, Elson & Baldwin, 1968). Closure of these linear oligomers into circles by the Escherichia coli DNA-joining enzyme was reported by Olivera, Scheffler & Lehman, 1968, for N 2 16. We report here measurement of the equilibrium constant for the minimum-size loop from the melting curves of open hairpin helices and also a study of the loop-weighting function, based on the melting curves of closed hairpin helices formed by circular oligomers. A study of loops in transfer RNA’s has recently been reported by Kallenbach (1968).

2. Analysis of Melting Curves (a) Possible confomndona In our previous paper (Scheffler et al., 1968) we discussed the formation of hairpin and straight-chain helices (dimers) from linear oligomers and the conditions under which each is formed. Here we wish to consider those portions of the melting curves which oorrespond to the helix-coil transitions of hairpin helices. An open hairpin helix can melt in three ways. by opening base pairs at the end, or by enlarging the loop, or by opening a second loop. These conformations are shown in Figure 1 together with “unsymmetrical” hairpins which have off-center loops. The longer non-bonded end of an unsymmetrical hairpin may terminate either with a 3’-OH group or with a 5’.P group; these are distinguishable classes (Scheffler et al., 1968). A closed hairpin helix can melt by enlarging either or both of the terminal loops, or by opening an interior loop. The totally melted form is a circular “random chain”. (b) Bake parameters (8, y, n). The partition functions are obtained by assigning relative statistical weights to the conformations shown in Figure 1. We follow the adaptation of the simple Ising model to the helix-coil transition of double-helical polynucleotides (for a review, see Applequist, 1967), but we interpret the nucleation parameter in terms which are somewhat different from those that others have used. Taking the totally melted random chain as the standard state (statistical weight equal to unity), an equilibrium constant 8 (the helix “stability constant”, Zimm, 1960)

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conformations

FIG. 1. Possible conformations of open hairpin helioes. Species I to III are included in the simple partition function, while species IV to VI are excluded. The effects of including species IV to VI are discussed in the Appendix.

is assigned to the reaction in which one base pair is added at the end of a helical stack -RT

Ins = AG = AH - TAS 0 = AH -T,

(1)

(co)AS.

(2)

Here AH, AS and AG are the standard enthalpy, entropy and Gibbs free energy changes that result from transferring one mole of base pairs from the random chain to the helical state. R is the gas constant, T is the absolute temperature, and T,( ao) is the melting temperature of poly d(AT), at which temperature the helical and random chain segments of the polymer have equal statistical weights. Nucleation of a hairpin helix involves closure of a loop with the formation of an isolated base pair. Loops of different sizes may be formed, with the condition that any loop must contain an equal number of A’s and T’s and therefore have an even number of non-bonded bases. We will relate the equilibrium constant for loop closure y(x) to the equilibrium constant K for forming an isolated base pair in an intermolecular reaotiont. (3) Y (4 = K P @). Here p(x) is the effective concentration (molecules/A3) of one base in the neighborhood of the complementary base and K has reciprocal units (8”/molecule). The loop size x t See the comparison by Wang & Davidson of half molecules of X DNA.

(1966) of the cyclization

reaction

with the joining

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is equal to the number of links, counted as internucleotide phosphate residues; there are x- 1 non-bonded bases plus one base pair in a hairpin loop. The standard free energy of loop formation will be written A(%,, (4 = - RT In y (x) = - RT ln K P (2). The function p(x) will be referred to as the loop-weighting function. In writing equation (3) we are departing from the usual formulation in two respects. In the conventional formulation, (1) the equilibrium constant for helix nucleation is written as the product of s times a nucleation parameter, and (2) the nucleation parameter is based on a comparison of loop formation with helix propagation (cf. Zimm, 1960; Applequist, 1967). The inclusion of s in the equilibrium constant for a lirst base pair gives compact equations and a decade ago it seemed physically reasonable first to count all the base pairs and then to include a factor allowing for differences between the properties of the first and succeeding base pairs. By now it is clear that an isolated base pair differs completely from a stacked base pair in its thermodynamic properties. In particular, the standard enthalpy of forming an isolated base pair is probably close to zero since hydrogen bonds to water must be broken to form the base pair. Consequently its equilibrium constant should be nearly independent of temperature while the stability constant s depends strongly on temperature. Therefore, it is confusing to include s as one factor of the equilibrium constant for an isolated base pair, and we now omit it. Our reason for expressing y(x) in terms of K and p(x) is that these should be independently measurable quantities for all except very small loops. In contrast, the standard comparison of loop initiation with helix propagation requires factoring s into two equilibrium constants, neither of which is measurable by the methods used in this work (see Discussion). The first reaction is the formation of an isolated base pair by closing a “loop” with only two links, and the second reaction is addition of this base pair to the helix, thus gaining a stacking interaction. In his original formulation of the theory, Zimm (1960) bypassed this problem by assigning a dellned value to AG,,,,(2) with the use of the Jacobson-Stockmayer 1.w.f.t for large loops. In this way a comparison between y(x) and s can be made in terms of measurable quantities (see Discussion) but the comparison loses much of its physical meaning, since it is based on dellned rather than actual free energy changes (cf. Crothers & Zimm, 1964; Applequist, 1967). Measurement of K from studies of dimer helices is straightforward if one has a simple experimental system (see the study of oligomeric rA: rA helices by Applequist & Damle, 1965) and we will not discuss it here. It is the equilibrium constant for joining two complementary strands by formation of a first base pair, and it has conventionally been written p s rather than K (cf. Applequist, 1967). The evaluation of p(x) has its origins in the pioneering study by Jacobson & Stockmayer (1950) of ring formation during synthesis of linear polymers. They computed p(x) for long chain polymers whose conformations follow a gaussian distribution, and they pointed out that the results could be expressed in terms of measurable quantities if one compares an intramolecular equilibrium constant for cyclization with a bimolecular equilibrium constant for joining two linear chains. Their treatment was extended by Flory & Semlyen (1966) to include short chains. t Abbreviation

used: l.w.f.,

loop-weighting

function.

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In either case p(x) is related to (T:), the mean-square end-to-end distance of a chain with x monomer residues, by P(X) =

[&)13’2w

(4)

where #(x) is defined below. Flory & Semlyen pointed out two factors which must be considered for short chains: (1) the proportionality between (rz) and x does not hold for short chains; and (2) the probability of loop closure will be affected if the ends of a short chain have a correlated angular orientation. Two conditions must be fulfilled for loop closure via a chemical reaction such as base pair formation. First, one base must be within a volume element 6v of the other, where the radius of &J is of the order of one bond length (Jacobson & Stockmayer, 1960). The second condition is that a line joining the bonding elements must fall within a critical solid angle w if bond formation is to occur. Let P,(W) be the probability that the relative orientation of the two bases is within the solid angle w when the second base is within the volume element 6v, and a loop of x units is to be closed. The corresponding probability for the bimolecular reaction is ~14~. Then

e4=

P*fW) (--q-q)’

(5)

For long chains P,,(W) becomes (w/4,) so that #(x) becomes 1, but for short chains p=(w) may be either larger or smaller than (w/4n) so that $(x) may be either larger or smaller than 1. For short chains, Flory $ Semlyen (1966) consider chain conformations in a 0 solvent (Flory, 1963) and define a characteristic ratio C, by the relation (r:) = XV PC,, (6) in which x is the number of monomer units in the chain, v is the number of rotatable skeletal bonds, and la is the mean square bond length. For long chains C, approaches the constant value C,. With the use of equations (4) and (6) the loop-weighting function for gaussian chains becomes P(X) = # (4

The Jacobson-Stockmayer C,/C, both equal 1.

J/bvz/~al)1”‘”

J = [3/2 T vZ~C,]~‘~. 1.w.f. for long chains is obtained by letting

17) (74 #(x) and

lim p(z) = J/x~/~. (8) =--*m Since the d(AT) chain conformations may not follow a gaussian distribution, we will also test an empirical 1.w.f. in which the exponent a is allowed to have some value larger than 312 (of. Fisher, 1966; Applequist, 1967) and in which C,/C, is replaced by a function f(z), which approaches 1 for large x.

p(x) = J’/[qfkW. Finally for gaussian chains we write the equilibrium with x links as

(9) constant for closure of a loop

Y(X)= tw M~~C,/Q~)1”‘” h= KJ.

(10) W)

All of these quantities except #(x) are obtainable from independent measurements, and #(x) becomes 1 as soon as the angular orientations of the bases closing the loop are

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uncorrelated before loop closure. The representation of CJC, is disoumed under Results. Because hairpin helices melt chiefly from the open end, the minimum-size loop predominates during melting (see the Appendix) and y(x) for this loop acquires an especial significance. We will denote it simply as y and use it as a basic parameter in the partition function. The third basic parameter is n, the maximum number of base pairs that can be formed in a hairpin helix from a linear oligomer of given molecular weight. If there are 2N nucleotides in the oligomer and g non-bonded bases in the minimum loop, then n = N - g/2. (11) (c) A simple partition function for open hairpins An approximate partition function, adequate for many purposes, is obtained by making the following assumptions. (1) The loop remains fixed at its minimum size during melting so that y(x) reduces to the constant y. (2) Conformations with 2 or more loops are excluded. (3) Any conformation with k base pairs is assigned the same statistical weight, ysk-l. The number of such conformations is 2(n-k) + 1. For an analysis of these assumptions, see the Appendix. The resulting partition function is P=l+y

i:?n,ak-i k=l

mk = 2(n-k)

+ 1

in which the random chain is assigned a statistical weight of 1. Evaluation gives P=l+(-5){(8~)Lsn+8”-~-2]-(2n-l)}.

(12) (1-W of the sum (13)

To relate the measured melting curve to a predicted curve, we first note that the fraction helix, 8, is B=

; 0

5 kf(k)

(14)

k-l

where j(k) is the fraction of molecules with k base pairs

j(k) = '9. Since the molar absorbance of a base pair in a helical stack depends on the length of the stack, we require an expression for Hy(k), the hypochromicity per base pair in a helix with k base pairs. We define &r(k) Hy(k) = 1 - (16) 4 where &(k) and E& are the molar absorbanoes at 260 rnp per AT pair of a helix with k base pairs and of a random chain polymer, respectively. The measured absorbance A(T) at a temperature T is normalized by dividing it by the value for the random chain at high temperatures, A,,,. A(T) A,,,(T) = A. (17) hi

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A theoretical treatment of the function Hy(k) has been given by Tinoco (cf. DeVoe $ Tinoco, 1962) but we assume, as proposed by Applequist (1967), that

=

HY@)

HY

w

(00)[1 - W91

where Hy( co) is the hypochromism of poly d(AT). This is the same as saying that the hypochromicity per base pair is proportional to the number of stacking interactions (k - 1) divided by the number of base pairs (k). We further assume that the molar absorbance for (A + T) in a random chain is independent of the size of the oligomer, and that the absorbance of (A + T) in a loop is the same as in the random chain form. It follows that

$ V(k) HY

&,V’) = 1 - ;

k 2

0

WI

(19)

where f(L) and Hy(k) are given by equations (15) and (18). (d) Ckrcukrr oligonzers The corresponding partition function for a closed hairpin helix, formed by a circular d(AT), oligomer, is given below. We include species which have either one loop (the completely open circle) or two hairpin loops joined by a helical segment. Consider the partly hypothetical equilibria shown in Figure 2. The circle closure reaction shown in (a) has an equilibrium constant (B/2c)p(2c) where B is a bimolecular equilibrium constant for joining two linear molecules by a phoaphodiester bond (see the analysis by Jacobson & Stockmayer, 1950) and 2c is the symmetry number of the circle. From a kinetic point of view, the factor 2c arises because the circle can open in 2c ways to give the linear chain?, which can close in only one way to give the circle. Next consider the sequence of reactions shown in (b). First the 3’ end of the linear 3’

9%

1,

(b)

2P

\l @-

5’ 1

sk-'Kp(2p+ll

3’

1, 1

s"'KP(2p+l)B,o(2q+1

FIQ. 2. Derivation of the statistical weights of closed hairpin helices. See text for en explanation. The statistical weight of each species shown is given underneath Each arrow shows the number of ways the reaction can occur. t For this hypothetical

equilibrium,

we allow cleavage to produce both 5’.A and 6’-T ends.

it.

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BALDWIN

chain closes a hairpin loop with 21, non-bonded bases and k base pairs. Since the 3’ end is at a fixed location this species can be formed in only one way and the reaction has an equilibrium constant of ske1~p(2p + 1). Next the 5’ end reacts with the 3’ end to form a phosphodiester bond, thus closing a second hairpin loop with 2q nonbonded bases (p + q = c - k). This second reaction, which can occur in only one way, has an equilibrium constant of Bp(2q + 1) ifp # q, so that the constant for the over-all reaction in (b) is skd1B~p(2p + l)p(2q + 1). If the two loops are of equal size, with 2 h bases each, then the final product has a symmetry number of 2 (it can open in two ways to give the intermediate species) and the over-all equilibrium constant is sk- ‘KB(p(2h

+ 1))“/2.

To interpret the circle melting curves we consider the reaction from a closed hairpin helix to an open circle and assign the open circle a statistical weight of 1. Then the statistical weights of the asymmetric and symmetric helical species are, respectively W p.p

55.~Sk-lKP(2P

=

+

1) P@l +

1)/P&)

q=c-k-p w,,,

(20) (204

= c Sk-1 ‘+@h

+ 1))2/~(2c)

2h = c - k.

(21) (214

The latter occurs only when c - k is even, To find g(k), the fraction of all circular molecules with k base pairs, it is necessary to count every conformation once and to count no conformation twice. For the asymmetric species we may apply the restriction p(q in summing over p to prevent counting the same conformation both as (p, q) and (q, p). However, it is more convenient to count all asymmetric species twice and later divide by 2; then each symmetric species can be included in this summation at twice its statistical weight. This gives the following expression for g(k) g(k) = ;ijg

in which Q, the partition

c,Yi;;2

P@P + 1) PC&l + 1)

(22)

function, is

Q = 1 + -f!t-

=ig 8-l

p(2c) k=l

c-kig’2

p(2p + 1) p(2q + 1).

(23)

p=9/2

The Cnal expression for the normalized melting curve of a closed hairpin helix is

A,,,(T) = 1 - f ;$; IcgW+4k),

(24)

where Hy(k) is given by equation (18) and g(k) by equation (22).

3. Materials and Methods (a) Linear and circular d( TA),

ol~onws

In a previous paper (Soheffler et a.l., 1968) we have already described the characterization of the d(TA), oligomers, and in another publication (Olivera et al., 1968) the preparation and characterization of circular d(AT), oligomers have been presented. However, a few mod&rations were made in the large-scale preparation of circular oligomers as follows. Poly d(AT) (Schachmann, Adler, Radding, Lehman & Kornberg, 1960) was digested in a controlled reaction (Elson & Jovin, 1969) to d(TA), oligomers with pancreatia DNase and the reaction was stopped while a relatively large proportion of longer oligomers

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(N>ZO) was still present. The digest was fractionated by preparative polyaorylamide gel electrophoresis (Elson & Jovin, 1969) with a gel containing 15% mrylamide and 1*50/b bis-acrylamide. It had previously been determined with analytical gels that under these conditions the leading band contained the unfractionated, smaller oligomers (N< 12), and more than 30 bands of fractionated d(TA)N,la oligomers were counted. One hundred and fifty micromoles of d(AT) digest were placed on the l&cm gel in a Biichler apparatus; 250 fractions were collected, but the resolution was not as good as expected on the basis of the experiments with snalytioal gels. Three or four fractions were pooled and molecular weight determinations were made at alkaline pH (Scheffler et al., 1968). The analysis on analytical polyruuylamide gels showed one or two adjacent heavy bands, but there was also some contamination by shorter material which might have travelled in the form of aggregates on the preparative gel. To make circles, typically 1 to 2 Foles (phosphate) of d(TA), oligomer in 2 ml. were dialyzed against buffer for treatment with the joining enzyme (Olivera t Lehman, 1967). The solutions were heated briefly in a boiling water bath and the following additions were made: 500 ~1. of glycerol, 50 4. of bovine plasma albumin (100 pg/ml.), 200 ~1. of NAD (1.3 m&r). The mixtures were incubated at 4O”C, each with 14 units of joining enzyme. After 3 hr the same quantity of NAD and seven units of enzyme were added, and the incubation was continued for another 3hr. The samples were then dialyzed (24 hr) in the cold against buffer (E-buffer) for treatment with exonuolease I (Lehman & Nussbaum, 1904); 200 units of exonuclease I were added to each, and the mixtures incubated at 37°C for 1 hr. Subsequently, the fractions were extracted 3 times with 3 ml. of phenol saturated with E-buffer, followed by extraction with ether and evaporation to dryness. Finally, the samples were taken up in water, dialyzed against 0.02 M-NaOH, 0.001 M-EDTA, and then against O-1 M-triethylammonium bicarbonate, pH 78. Samples of these stock solutions were dialyzed against the required buffers for later experiments. The over-all yield of circles from the linear species was around 50%. Molecular weight determirmtions were again made in alkaline and neutral buffers, and a summary of these determinations is presented in Tablel. Judging from the melting curves (to be presented later) the contamination by linear material was less than 2%. TABLE 1

Molecular weig?ds of circuhr d (A T) oligomers Sizes of starting material (linear oligomers)

dCW,,.,t dCWm.,t t Determined by equilibrium sedimentation given in Fig. 1 of Scheffler et al. (1968).

Sizes of circular oligomers formed

d@‘h.at WTh.oS WW,,.,§ WW,,.,t at pH 12, 0.52 M-N~+ , using the calibration

curve

$ Determined at pH 7.0, 0.51 M-Ne+. $ Determined et pH 7.0, !Z 0.01 M-Ne+ . (b) Enzyme.3 A highly purified preparation of Eschmichia coli polynucleotide joining enzyme was generously provided by Dr Y. Anraku, who also recommended the addition of glycerol to the reaction mixture. Highly purified E. coli exonuclease I was a gift from Dr I. R. Lehman. (c) Mea.od8 The polyacrylamide gel electrophoresis (Elson & Jovin, 1969), the measurement of the molecular weights with the analytical ultracentrifuge, the determination of the melting curves of these oligomers and other methods have been described previously (Schefflel et d, 1968).

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4. Results (a) Melting curves for open hairpin ?&ice8 (i) Size of the minimum loop Model building with space-filling (CPK) models shows that the minimum loop need contain no more than four unpaired bases (g = 4). With CPK models a loop can also be made with only two unpaired bases but some of the contacts are tight. Because of the alternating AT sequence, the loop must contain an even number of bases. Our hypochromicity results for open hairpins are consistent with g = 4 (see Fig. 3); since the molar absorbance per AT pair of a hairpin helix is an average of those in the loop and in the helix the hypochromicity of short hairpin helices depends sensitively on g.

Fm. 3. Percent* different count&on A chain)) x 100 where random chain. Filled results measured on (Buffer compositions curves (see text) are bonded bases in the

hypoohromioity of open hsirpin helices es a function of oligomer size at three concentrcttions. The yc hypochromicity of d(TA), is defined as ( 1-(Ahsux/ A is the absorbance at 262 rnp of the complete helix or the totally melted symbols show the results measured on cooling while open symbols show the heating. Circles: < 0.01 M-Na+, triangles: 0.06 M-N&+, squares 0.61 M-Na+. are given in the preceding paper, Scheffler et al., 1968.) The theoretical (g = 2, 4 and 6) for the number of nonc&ulated for three different dues minimum-size loop.

(ii) Hypochromicity. Figure 3 shows a plot of the total hypochromicity as a function of chain length of each oligomer, measured at various oounterion concentrations. For hairpin helices there are two reasons for this dependence on chain length: (a) the hypochromicity of a base pair in a short helical stack depends on the length of the stack, and (b) there are g unpaired bases in the loop. The predicted curves in Figure 3 are calculated as follows. (1) Equation (18) (Applequist’s approximation) gives the dependence of hypochromicity on helix length. (2) The molar absorbance of (A + T) in the loop is the same as in the random chain. (3) The molar absorbance of (A+T) in the random chain is independent of chain length, counterion concentration, and temperature. (4) g = 2, 4 or 6. Only the curve for g = 4 fits the results. Other studies (Inman & Baldwin, 1962, and unpublished work) have shown that the hypochromicity of the random chain form of poly d(AT) is essentially independent

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of counterion concentration and not very dependent on temperature although the random chain is strongly hypochromic relative to its constituent mononucleotides. Probably the hypochromicity of the random chain is due to base stacking (possibly AA stacks) and we do not yet know why it shows only a small dependence on temperature. The hypochromicity measurements are taken from the cooling curves to minimize effects of dimer formation. At the lowest counterion concentration, where dimers have not been observed, measurements from the heating curves are also included. The predicted curve of hypochromicity (for g = 4) versus chain length agrees with the experiment except for one result, that for the shortest oligomer. This one result may be due to dimer formation, since both the rate and extent of dimer formation increase rapidly with decreasing chain length: Schefller et al., 1968. At the highest counterion concentration, where effects of dimer formation are observed in the time needed to measure a cooling curve, only results obtained by heating and quick cooling are included. (iii) Measurerneti of 8 aruEy. The stability constant s is calculated from values of AH (measured calorimetrically by SchefIler & Sturtevant, 1969) and AS (obtained from T, of poly d(AT) and AH: see equations (1) and (2)). The calorimetric measurements were made at a counterion concentration where the T, of poly d(AT) is near 40°C; they gave AH = - 7.8 & O-2 kcal./mole base pairs, and AS = - 24.8, e.u./mole base pairs. To relate these values to the present conditions (051 M-Na+, T, of poly d(AT) = 69.2%) we assume that the dependence of AS on salt concentration is probably small and can be neglected (cf. Krakauer & Sturtevant, 1968) and that AH can be calculated from equation (2) with AS = - 24.8, e.u. and T,( co) = (69.2 + 273.1,). This gives AH = - 8*5kcal./ mole base pairs. The calorimetric measurements show that AC, is very small, so that it is reasonable to treat AH as a constant at a given counterion concentration. The T, of an open hairpin depends on chain length only if y # 1, and the extent of the dependence of T,(n) on (l/n) is determined chiefly by the magnitude of y. The most direct way of evaluating y is from this dependence and the measurement can be checked by using y to predict the breadth and shape of the different oligomer melting curves. To find y we compare the experimental plot of T,(n) versus (I/n) (Fig. 4) with predicted curves obtained from computer-generated melting curves, using equation (19). To take account of dimer formation at low temperatures (Scheffler et aE., 1968), .we find the T, by subtracting from the top of the melting curve one-half of the hypochromism measured at the lowest counterion concentration where no dimers are formed. (Specifically, the curve for g = 4 in Figure 3 is used, which passes through these points.) The best value of y (equation (8a)) appears to be 0903 and the melting curves can be fit equally well with y = 0.003 or 0.004. If y is changed by a factor of two there are obvious differences between the predicted and experimental T, values, and also between the observed and predicted melting curves. (iv) Comparison of experimental and predicted melti~ng curves In addition to y, s and n, the predicted melting curves depend on the measured hypochromicity of poly d(AT), on g, and on the assumptions relating hypochromicity to helix content. As judged by Figure 3, these last assumptions are not a serious source of error. Since the T, valuss have already been used to determine y, one can test only the predicted breadth and shape of each melting curve. Figure 5 indicates that

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the agreement is satisfactory. In addition to checking the value of y, this comparison tests the self-consistency of the analysis and the adequacy of the simple partition function. One could obtain a more precise experimental test if there were no dimers formed at low temperatures. A study (Appendix) of the species omitted from the simple partition function shows that these have a slight but observable effect on the breadth of an oligomer melting curve, but little effect on its T,.

FIG. 4. Determination of y, the equilibrium constant for the minimum-size loop, from the dependence of the melting temperature on chain length; 7t is the maximum number of base pairs in a given hairpin helix. T,(n) is the temperature at the midpoint of the change in absorbance due to hairpin melting.

O”t , 20, , 40, , 60, , 80, , 100, 1 0

Temperature

03

* FIG. 6. Comparison of the observed melting curves for open hairpin helices with curvea predicted from the simple partition function (see also Fig. 11). Open symbols show the heating curve*, filled symbols show the cooling curves. At this high counterion concentration (0.51 M-NE+ ) dimer helices are formed at low temperatures. At lower counterion concentrations where dimers are not formed this partition function is inadequate because of electrostatic effects which vary with chain length.

HELIX

FORMATION

It should be emphasized that ment only at high counterion static effects can be neglected. low counterion concentrations (v) Helix formation

BY

d(TA)

OLIGOMERS.

II

157

equation (19) gives satisfactory agreement with expericoncentrations, where chain-length dependent electroThe behavior of these oligomers at intermediate and will be discussed in a following paper.

by d( TA),

Helix formation at room temperature has not been observed in short dimeric helices such as (rA),:(rA), at acid pH when 11is four or less (Applequist & Damlc, 1965). Consequently, it is interesting to observe that d(TA)5 does show a melting curve (Fig. 6). Like the larger oligomem d(TA), gives a biphasic melting curve at 0.51 M-Na+, but the first traneition now makes the major contribution to the absorbance change. The upper part of the melting curve suggests partial formation of hairpin helices which melt at high temperatures. Since the minimum loop must have at least two and more likely four non-bonded bases, the complete hairpin helix formed by d(TA), could have three or at the most four base pairs.

:L-I-l--

IO 20

30

40

50 60

70

80 90

Temperature (“C ) Fra. 6. The biphaaio transition curve of d(TA), at 0.51 w-X8’, comp8red with the transition curves of longer oligomere. The upper portion of the curve for d(TA)6 auggeate partial formation of heirpin helicea at temperaturea above 60°C. These could have no more than 3 or at the moet 4 beee pairs.

(b) Melting of clod

hairpin

helices

The melting curves of two circular oligomem at two different salt concentrations are shown in Figure 7 together with the melting curve of poly d(AT). The increased stability of the closed hairpin helices relative to poly d(AT) is very striking, particularly at the lower counterion concentration, and it should also be noted that the smaller of the two oligomers is more stable under all conditions. Since a closed hairpin helix has two minimum-size loops, the measured hypochromicity is smaller than that of open hairpins of corresponding size. Using the same assumptions as those presented in the discussion of the hypochromicity of open hairpins one can calculate the minimum loop size, and from the measured value of 26.5% for d(AT),,,, one obtains g = 2.6; from the larger circles a slightly smaller value of g is calculated. There appears to be a small discrepancy between the loop size calculated from circles and from hairpins. The data for open hairpins suffer from an uncertainty in extrapolating the baselines of the melting curves while the circle

168

I.

E.

SCHEFFLER,

E.

L. ELSON

-4ND

R. L.

BALDWIN

-!

0.80

IO

20

30 40

so

60

70

80

Temperature (‘Cl

60

70

Temperature

80

90

K 1

FIG. 7. The melting curves shown by two circular dAT oligomers tration (< 0.01 M-Ne+) 8nd (b) at 8 high counterion concentration

(8) St 8 low counterion (0.51 M-N&+).

concen-

data were obtained with slightly inhomogeneous preparations. In predicting the melting curves for circles we continue to use g = 4 for the minimum size of a loop. Study of t?beloop-weig?&ng junction Some insight into the physical factors which control the transition from open circle to closed hairpin helix is provided by first treating the reaction as all-or-none. Then one may set the statistioal weight of the complete helix equal to one at the observed T, 1 = 8 - 5w

PWPW

W)

and, since c and y are known and s can be computed from T,(c) - T,(m) (see equations (1) and (2)), we may compute r = p(5)/p(2c) and compare it with the value given by the Jacobson-Stockmayer 1.w.f. For c = 20, T,(c) - T,(a) = 4+F’C and r = 226 compared to r (Jaoobson-Stockmayer) = 22.7; for c = 31, T,(c) - T,( 00) = 4.0°C and Y = 450 compared to T (Jacobson-Stockmayer) = 43.9.

HELIX

FORMATION

BY

d(TA)

OLIBOMERS.

II

169

The comparison indicates that the Jacobson-Stockmayer 1.w.f. is not adequate to represent the circle melting curves. The difference between the observed and the Jacobaon-Stockmayer values for r might result either from omitting excluded volume effects (which affect the exponent a) or from omitting the dependence of C, on x (which affects chiefly the values of p(5)). We ask first whether increasing the value of a can account for the results. Taking p(5)/~(2c) to be (2~/5)~, we find that even when u is increased from 1.5 to 2-O the predicted values of r are increased only to 64 (c = 20) and to 153.8 (c = 31). We conclude that excluded volume effects are not sufficient to account for our results. Next we ask whether changing p(5) to a smaller value can explain the results; we take p(5)/p(2c) to be (p(5)/J) (2~)~‘~ and compute p(S)/J from the observed values of T for c = 20 and 31. If this is the correct explanation, we should obtain the same p(5)/J for both values of r and it should be considerably less than (1/5)3’a = l/11*2. The values of p(5)/J computed in this way are l/O99 for c = 20 and l/O.92 for c = 31. This close agreement indicates that the dependence of C, on x is a principal factor in the interpretation of the circle melting curves. To carry the analysis further we must use the complete partition function (in equation (24)) since representation of the transition as an all-or-none reaction gives melting curves which are too sharp although their T, values are predicted fairly well. Figure 8 shows that, when the complete partition function is used, the JacobsonStockmayer 1.w.f. fails to predict the observed T, values and that increasing the exponent a in p(x) = J1/xa to a = 2 or even a = 3 fails t,o predict the observed behavior.

Temperature (“C ) FIG. 8. The observed melting curves for circular oligomers are compared with ones predicted by UBBof the Jacobson-Stockmayer loop-weighting function (a = 3/2) and with other curves in which a is allowed to have values larger than 3/2. No single value of a gives a satisfactory fit. (4 Poly d(AT)<,,>. (b) Poly WT).

A minimum representation of the dependence of C, on x is given by using a step function: we allow p(5) to have a special value and UIX the Jacobson-Stockmayer 1.w.f. for all loops larger than five links. The predicted melting curves are baaed on equation (24) and the complete partition function. As measured by curve-fitting this works

160

I.

E. SCHEFFLER,

E. L. ELSON

AND

R. L.

BALDWIN

quite well: the fit of the melting curves with p(5) set equal to J/16 is as good as any obtained with l.w.f.‘s which are physically more reasonable. The study by Flory & Semlyen (1960) of C, veraus x for poly (dimethylsiloxane) provides a useful test of possible empirical representations. Their curve shows the following properties: (1) for z = 1, C, = 1, (2) for small values of x (giving (C2/Cm) < 0.7) the function (1 - eekrZ) represents (C,/C,) fairly well; and (3) for large values of x (with (C,/C,) 2 0.6) the function (C,/C,) = 1 - (k/x) fits quite well. Both of these functions have been used to predict the circle melting curves, with k or k treated as an adjustable parameter, and both work reasonably well (using equations (7), (22) and (24)). The curves predicted for c = 20 and 31 with (C,/C,) = 1 - k/x are shown in Figure 9 with k = 4.3. When (C&Y,) is represented by (1 - ebk’+) the 6t is somewhat less good and the best fit is obtained with k’s near 0.03. Our results do not distinguish between the validity of a = l-5 and a = l-75; if a is taken as l-75 the circle melting curves can still be fitted satisfactorily by using a smaller value of k.

Temperature

C’C)

FIG. 9. The melting curves for circular oligomers which are predicted when the departure from random-flight behavior of short ah&a is taken into account by equation (7) with $ (5) = 1 end ‘A/C, = l-(4*3/4. (a) dW). (b) WW.

5. Discussion (a) The loop-weighting function and the equilibrium constant for loop closure Our results may be summarized as follows. (1) The equilibrium constant for closing the minimum-size loop in a d(AT) hairpin helix is y = O-003 f 0901; y is taken to be a constant independent of temperature and the number of non-bonded bases in the minimum loop is assumed to be four (see Results). (2) The Jacobson-Stockmayer loop-weighting function fails for short d(AT) chains, probably because the characteristic ratio C, drops sharply as x approaches 1. Since C, variest from a value near 1 t When the epan of the repeat unit is tied, C, = 1 when z = 1; for example in polypeptides C, = 1 at z = 1 if the repeating unit is the C~--C~ distance. For polynucleotides the span of the repeat unit is expected to be variable, and to depend on the local conformation (P. J. Flory, personal communication, 1969). Nevertheless, C, is expected to drop in a similar fashion aa z approaches 1.

HELIX

FORMATION

BY d(fA)

OL~GO~E~S.

11

Ml

at x = 1 to C, at large x, the size of the drop depends on C, : poly d(AT) in the random chain form may well have C, greater than 10 (see below). (3) The equilibrium constant for closing a d(AT) hairpin loop with x links (and x - 1 non-bonded bases) can be represented within the accuracy of our experiments either by an equation derived for gaussian chains (264 Y&J = uw*/~ax’” or by a similar equation based on the Fisher (1966) approximation, Y(4 = ~lkfi~))‘*‘”

(26b)

These equations assume that the orientation factor 4(z) (equation (5)) is unity for all values of x being considered. (4) Because the predicted melting curves for circular oligomers depend strongly on the change in C, with x for small loops, the experimental ourves cannot be used at present to measure excluded volume effects (i.e. to determine precisely the exponent a of the 1.w.f.). Since our results are compatible with at least three different empirical representations of C,/C, (see Results), we will give the values of h determined for each. (a) The step function (C&Y, = 1 for 5 b 7, C&Y, = 0.26 for z = 5) gives X = 09045; (b) a function which is capable of fitting well at large x (C,/C, = 1 - 4*3/x for x 2 5) gives X = 0902; and (c) a function which is able to fit well at small x (C&Y, = 1 exp ( - 0.032)) gives /\ = 0902. When an exponent of 1.76 is used in the 1.w.f. (equation (26b)), the best fit with f(x) = 1 -k/x is obtained when k = 3.8, and this gives A’ = 0902. Although all these representations of C,/C, give values of X in the range 0902 to 0.005, other allowable representations might give values outside this range. Further analysis is deferred until C, and the curve of C, veTsuBx are determined from studies of chain conformation. Our estimate of h may be compared with the related parameter r, determined by Crothers & Zimm (1964). They analyzed the data of Inman & Baldwin (1964) for the co-operativity of melting shown by the DNA homopolymer pairs poly dI*dC and poly dI*dE, by use of a theory for the melting curves of DNA polymers with repeating sequences. Their theory employed the Jacobson-Stockmayer 1.w.f. and may therefore be subject to future modification. The relation between r and h is 7-l

=

X/s3"8.

(27) It is obtained as follows. Zimm (1960) separated AG, the standard free energy of base pair formation, into two terms AG = AC;,,

(2) + AG;$

(28) where AC ioop(2) is by definition computed from the Jaoobson-Stockmayer 1.w.f. for a “loop” containing two links (i.e. two phosphate residues and no non-bonded bases) and AC’,, is whatever remains. If we use equation (8) for the Jacobson-Stockmayer 1.w.f. together with equation (3a), we have (2) = -$ (29) > where J is defined by equation (7a). Since 7 is defined (Crothers & Zimm, 1964) as exp

equation (27) follows directly. 11

(

$100~

162

1. E.

SCHEFFLER,

E. L. ELSON

AND

R. L. BALDWIN

From our results and equation (27) we compute that p-l is of the order of lo-” to 4X10-* ifs = 2 (a typical value in our experiments) and A = 0.002 to 0.005. This agrees with the estimate by Crothers $Zimm (1964) of 7-l = 10-s to 10m4.However, X may have different values for different DNApolymers since h = KJ (equation (IOa)) and K may depend on the type of base pair formed while J will depend on C, (see equation (7a) and below). The object of expressing y(z) in terms of K and p(s) is to compare the observed value of A with that predicted from h = KJ, using measurements of K and C,, We have a preliminary value for K, based on an analysis (Elson, results to be published) of the temperature-dependent equilibrium between monomer and dimer d(TA), helices (Scheffler et al., 1968): Kg 7A3/molecule. This is similar to the value of K for rA:rA oligomeric helices formed at acid pH. Applequist & Damle (1965) report /I = 2.2 X low3 liters/mole in studies where a typical value of s is near 6; this gives K = p.3 = 1.3 X10-e liters/mole or 22 A3/molecule. The value of C, for poly d(AT) has not yet been determined. Results for several other polynucleotide chains have been obtained by Felsenfeld and co-workers (cf. Eisenberg & Felsenfeld, 1967); using an extrapolation to remove the effects of base stacking, they find values of C, in the range of 17 for certain polyribonucleotides including poly rA and poly rU (G. Felsenfeld, personal communication, 1968). Analysis of a particular model for the unstacked polyribonucleotide chain by Wilma King and P. J. Flory (personal communication, 1969) has yielded a curve of C, versus x with a smaller value of C, and shows a strong dependence of C, on the type and extent of base stacking. The bases in the random chain form of d(AT) probably are stacked in some manner, since poly d(AT) shows a large hypochromicity relative to the mononucleotides (Inman & Baldwin, 1962). For illustration we compute X for G, = 20 and for K = 7A3/molecule: h = 0.5 X 10w3. This is to be compared with our measured value of A, 2 to 5 X 10e3. At this stage of the analysis, the agreement is reasonable. It should be pointed out that we treat the 1.w.f. as being dependent only on loop size and not on temperature. Since the 1.w.f. is expected to depend on the degree of base stacking in the random chain, it may vary with temperature as well as with the type of polynucleotide chain. (b) The stacking interaction in single-strand and double-strand helicw We wish to discuss here some examples which illustrate the physical origins of the co-operativity of polynucleotide helix-random chain transitions. Consider first the effects of breaking a staoking interaction in a single-strand helix such as poly rA at neutral pH, which has been studied by many workers (for a review see Felsenfeld & Miles, 1967). Base stacking in poly rA is non-co-operative or only weakly co-operative, and the standard enthalpy per stacking interaction appears to be at least as large as that for stacking base pairs in DNA. To a good approximation, the stacking interaction depends only on nearest neighbors so that a stack of k bases can be described by k - 1 equivalent stacking interactions. Consequently helix melting is non-co-operative because the result of removing one base from the end of a stack is the same as that of parting the stack in the middle (see Fig. 10) except for possible rotation about the glycosidic bond of the single unstacked base in case (a). In both cases the number of stacking interactions is reduced by one. If helix melting were to be co-operative, removal of a base from one end of a helical stack would be favored over breaking the stack in the middle.

HELIX

FORMATION

Single strand helix

BY

d(TA)

OLIQOMERS.

Double

II

163

helix

FIG. 10. A diagram illustrating the relation between the co-operativity of melting and breaking a stacking interaction at the end or in the middle of a helix (see text). For the double helix the stacking interaction is supposed to be broken by extending the backbone, in the manner often postulated to allow intercalation.

Consider next the theoretical case of a two-stranded monomer : polymer helix. Addition or removal of a monomer is a co-operative process because one stacking interaction is lost if a monomer is removed from the end of a stack while two stacking interactions are lost if the monomer is taken from the middle of a stack (cf. Magee, Gibbs & Zimm, 1963). In principle the helix could also melt merely by breaking stacking interactions, and this by itself is non-co-operative as in the case of the single-strand helix. However, when an isolated base pair is formed by breaking a stacking interaction at the end of a helix, the monomer dissociates. This favors breaking the helix at an end, making this also a co-operative mode of melting. When a stacking interaction is broken in a DNA helix by extending the phosphodiester backbone, just one stacking interaction is lost whether the helix is parted in the middle or at an end: see Figure 10. The loss of a stacking interaction is not in itself sufficient to cause co-operative melting of a DNA helix, since the conformations shown in (a) and (b) have equal numbers of stacking interactions, and of base pairs, and also have loops of equal size. Because the isolated base pair in (a) can open to allow unconstrained rotation of the phosphodiester bonds, while the interior loop in (b) constrains the rotation of these bonds, the over-all reaction shown in (a) will be favored. Consequently, the removal of a loop can be sufficient to cause the co-operative melting of a DNA helix. To compare the equilibrium constants for initiation and propagation of the DNA helix, we need to consider the following factors. For initiation we have seen that the equilibrium constant can be written as I++) (loop formation: see equation (3)) or simply K (joining two strands). For propagation we divide the process of forming a stacked base pair into two hypothetical (but possible) steps: first an isolated base pair is formed, and next it is stacked at the end of the helix. This is the reverse of the two reactions shown in Figure IO(a). Then we may write s = K P(2)T’

(31) where p(2) is the actual (not the JacobsonStockmayer) 1.w.f. for a loop of two links and T’ is the equilibrium constant for the DNA stacking interaction shown in Figure 10. Unfortunately, p(2) cannot be evaluated by the methods used in this work: the 11’

164

I. E. SCHEFFLHH,

I<:. 1,. I~LSOS

.\SI,

I:.

I,.

I{.\l,i,\\IS

orientation factor #(2) (see equat’ion (1)) is 11rob;ibly quit~c different. from OII(‘, iltlll none of our approximations to the curve of C,/f’,, ??erstLs IUwould give a reliable value, for C; when 2 = 2. Our best estimate of p (2) is obtained by letting CT = I since C, is close to 1 when 1: = 1 (see footnote on p. 160); then p (2) = J(C,/1”)s/2 and 7’ = 0.014 T if C, = 17. One way of comparing s with y(z) is to combine T’ and p(Z) into a single measurable quantity, w. w = s-/p(S) (32) which gives S=KW.

Wa)

Then the ratio of y(x) to s is simply Y(X) s

PC-4 w

>

(33)

where p(s) is given by equation (7) or (9) and w is expected to have a strong dependence on temperature, like s. If K is known, w can be computed from s. In principle p(x) can be found from studies of chain conformation and y(x) from measurements of helix-random chain transitions. In discussing different stacking interactions, it is important to note that the strength of the stacking interaction depends on the extent to which the favorable change in enthalpy is compensated by an unfavorable change in entropy. The net stacking interaction is often a weak interaction, as in the study by Davis $ Tinoco (1968) of base stacking in ribose dinucleoside phosphates for which base stacking is accompanied by a loss of free rotation about phosphodiester bonds as is also the case in DNA. In these compounds the equilibrium constant for stacking is of the order of magnitude unity even at low temperatures. Of the 16 possible dinucleosidephosphates, all but GpG were studied. The average values of AH and AS were found to be -6.5% 1.5 kcal. and - 25 & 5 e.u. per stacking interaction. (These numbers are approximate since they were measured from van’t Hoff plots and Davis & Tinoco found that base stacking in such compounds is not a two-state reaction. Similar values of AH and AS have been reported by Brahms, Maurizot & Michelson (1967).) They may be compared with the ones given here for formation of a stacked base pair in poly d(AT) : AH = - 7.8 kcal. and AS = - 24.8 e.u. per mole of base pairs in O-01 M-N&+. In both cases a large negative standard entropy largely compensates the favorable change in enthalpy on stacking. In the case of the defined stacking free energy AGISt (equation (28)) evaluated by Crothers & Zimm (1964), the total free energy change AG for forming a stacked base pair was separated into two terms, one of which (AG’,,,,(2)) COITeSpOndS approximately to one-half of the total standard entropy for dAT of - 25 e.u. (this can be calculated by equation (29)), 1eaving only the remaining half to partially compensate the standard enthalpy. The result is a large (but arbitrarily defined) stacking interaction: see the discussion by Crothers & Zimm (1964) and Applequist (1967). Our discussion of helix initiation has focussed on the formation of isolated base pairs, in a comparison of the intramolecular and bimolecular reactions. It should be pointed out that there is as yet no proof that initiation of the DNA helix occurs via the formation of an isolated base pair. The actual mechanism might involve the simultaneous formation of two or three stacked base pairs from complementary stacks of single bases. We have referred to K as the equilibrium constant for an

HELIX

FORMATION

BY

d(TA)

OLIGOMERS.

II

166

isolated base pair but it could also be described as the equilibrium constant for the first base pair in a helical stack of k base pairs, where the combined equilibrium constant for the other (k - 1) base pairs is 6-l. The work of Tuppy & Kiichler (1964) indicates that complementary nucleosides do form isolated base pairs in aqueous solutions. They made chromatographic columns in which the ribonucleoside of A or G was covalently bound to the column, and showed that these columns would separate mixtures of the ribonucleosides of C and U: C was retarded if the column contained G while U was retarded if the column contained A. (c) Comparison of base stacking with “hydrophobic bonding” in proteins Studies of model compounds indicate that both proteins and nucleic acids owe much of their stability to the formation of structures which exclude water. Nevertheless the characteristic values of AH and AS are opposite in these two cases. Close packing of the hydrophobic side chains in proteins is an entropy-driven reaction in which enthalpy changes play only a minor role (cf. Kauzmann, 1959). Base stacking in nucleic acids is an enthalpy-driven reaction which is opposed by an unfavorable change in entropy. If, as is often supposed, both processes are actually driven by a rearrangement of the water structure, then it would appear that water forms characteristically different structures around the purine and pyrimidine bases as compared to the hydrophobic side chains of proteins. In fact, the physical reasons why base stacking occurs are still undecided: see the review by Felsenfeld & Miles (1967). There is a second type of evidence that base stacking and protein side chain packing differ characteristically in their thermodynamic behavior. The unfolding of a protein shows a large and temperature-dependent difference in heat capacity, AC,: see the data of Danforth, Krakauer $ Sturtevant (1967) for bovine ribonuclease A and the analysis of this unfolding reaction by Brandts & Hunt (1967). However, the melting of poly d(AT) shows a AC, which is too small to measure reliably (Scheffler & Sturtevant, 1969).

APPENDIX

The model for the melting of hairpin helices as described above is simplified by neglecting avariety of possible but presumably improbable conformations. In particular, we have assumed that a hairpin helix has only a single loop and that this loop does not enlarge as the molecule melts. The melting is supposed to occur entirely at the open end of the helix. Hence, only species of types I, II and III in Figure 1 are counted. The purpose of this Appendix is to test these assumptions by evaluating the contribution of the neglected conformations. This more complete treatment accounts for conformations in which (1) there is a single hairpin helix with loop of variable size (species II, III and IV in Fig. l), (2) there are two hairpin helices with loops of variable size (species VI in Fig. l), and (3) there is an interior loop as well as the hairpin loop, both of variable size (species V in Fig. 1). The statistical weight of the completely melted form is unity. Hence, the sum of the statistical weights of all allowed conformations is & = 1 + Q1 + Qa + Q3. We evaluate these partition functions for an oligomer of 2iV nucleotides, d(TA),. For an interior loop we assume that the hairpin loop-weighting function can be used; an interior loop with x links (or phosphate residues) contains x - 2 non-bonded bases.

166

I.

E. SCHEFFLER,

E. L. ELSON

APL‘D It.

I,.

BALDWIN

(1) Single hairpins with variable loops In conformations having k helical base pairs and 21nucleotides in the loop there are 2(N - k - 1) + 1 ways to distribute the “linear” non-bonded nucleotides (those not in the loop) between the 3’- and 5’-ended portions. The statistical weight of each of these conformations is Sk-‘y(21 + 1) = K,,(21 + l)Sk-'. The sum of the statistical therefore :

weights of all conformations with k helical base pairs is N-k

- k - 0 + ~IPW + 1)).

F,(k) = K csk-l ,&{[W Hence the partition

(A-1)

function is: N-da

&I =

C

(A-la)

P,(k).

k=l

(2) Double hairpins In general a double hairpin can contain seven conformational regions: two helices, the two accompanying loops, and three linear regions of non-bonded nucleotides (species VI(a), Fig. 1). The last, any or all of which can contain zero nucleotides in particular examples, include the 3’- and 5’-ended portions and the region separating the two helices. We imagine the double hairpin to be divided into two single hairpins. For definiteness one hairpin may contain the 5’-end, the adjacent helix and loop, and the interior non-bonded portion. The other, then, along with the adjacent helix and loop has the 3’-end as its only linear region of non-bonded nucleotides. The statistical weight of any double hairpin conformation is the product of the statistical weights of the appropriate single hairpin conformations. Suppose that 2A nuoleotides are allotted to the first hairpin and 2B = 2(N - A) to the second. The sums of the statistical weights of conformations of the first and second hairpin which have k and j helical base pairs respectively are: A-k

T,(A, k) =

K 8’-’

l~,a{W4

and:

- k - 1)+ LIP@ + 1))

(A-2a)

E-i T,(B, j) =

KS’-’

2 1=9/a

~1%

f

1).

(A-2b)

The statistical weight of a double hairpin composed of these two single hairpins, one with k, the other with j helical base pairs is: The partition

FLA’ (k, j) = T,(A, k) T,(B, j). function is given as: N-(g/2+1)

A-g/2

(A-3)

N-A-g/1

(A-3a) A stacking interaction may develop between base pairs formed by the 5’-terminal thymidylate and the 3’-terminal adenylate residues when these are at the ends of separate helical stacks (see species VI(b), Fig. 1). We may write an equilibrium

HELIX

FORMATION

BY

d(TA)

OLIGOMERS.

II

167

constant 78 for this staoking interaction. The unstacked conformation has a statistical k - 1) + 1) if there are a total of k helical base weight of ~%“-~p(2E f l)p(2(N nucleotides in the two loops. The stacked conforpairs with 21 and 2 (N - k - 2) mation has this statistical weight multiplied by 7*. As a probable upper bound to 7Swe may use the parameter T’ of page 164 and, in accordance with the discussion on page 164, we take T’ = 0014, T = 40 (see equation (27)). There are [k/2] adenylate residues to which the terminal thymidylate may be paired. Here [k/2] is the integral part of k/2. Hence the sum of the statistical weights of all conformations with k helical base pairs which are stabilized by the additional interaction is: N-k-PI2

1 ~(21 + 1) PPW -- k - 1)+ 11. (A4) 1=0/a The sum of the statistical weights of all conformations in which the interaction occurs is: N-g c = 2 F,(k). (A-5) F,(k) = rs

KS

[k/2] sk-2

k-a

This sum replaces the sum of the corresponding terms in Qa in which the interaction is omitted. This latter sum can be written as (C/T*). Hence, the additional stacking interaction is accounted for by adding to Qa the term: Q, = C (1 - l/r,). (A-6) (3) Hairpins with interior loops The following four observations define the probability of occurrence of hairpin conformations with an interior loop (species V in Fig. 1). (a) A conformation with a total of k helical base pairs has k - 1 positions for the interior loop. (b) An interior loop with 21, nucleotides can be formed in (21, + 1) ways depending on the number of nucleotides contributed by each of the two complementary strands (cf. Crothers Q Zimm, 1964). (c) These are 2(N-k-2, -la) linear non-bonded nucleotides in a hairpin with k helical base pairs and 2Z1and 21, nucleotides in the hairpin and interior loops respectively. These linear residues can be distributed in 2(N - k - 1, - ZJ + 1 ways between the 3’- and 5’-terminal portions. (d) The 1.w.f. for an interior loop of 21, nucleotides is ~(21, + 2). In the light of these postulates the sum of the statistical weights of conformations with an interior loop which have k helical base pairs is: N-k-l

F,(k) =

.a Ik -

K=#-

1)

C ~(24 + 1) II = !?/a N-k$,

II

(PO’

- k - 4 - 41)+ 11PC%

+ 2))

(A-7)

and N-tr,2+1)

Qs=

2

Pa(k).

(A-7a)

k=l

III order to compute results comparable to experimental observations it remains to express the contribution of the various oonformations to the relative absorbance. A

I. E.

168

conformation contributes

SCHEFFLER,

of statistical

E. L. ELSON

AND

R. L.

BALDWIN

weight P(k) with a single stack of k helical base pairs

Hence, the part of the total relative absorbance due to single hairpins with one loop is: A, = $ .z” k

F,(k) [l 1

;

Hy(k)] = 2 - &

N$ia U’,(k)&(k). k

0

(A-8)

1

By the same line of reasoning the relative absorbance due to double hairpins is: 1 N-@/2+1) A-g/2 B-g/Z c 2 e%j@ (X) HY(k) - (f) HY(j)l k=l

j=l

= Qa,Q

,zl

jzl GYk

+ jfwj)l.

j)WY(k)

(A-9)

This expression may be corrected to account for the stacking interaction between adjacent 3’- and 5’-terminal helical base pairs. The contribution to the relative absorbance of a conformation of statistical weight F(k, j) in which a total of k helical base pairs is divided into two stacks of j and k - j base pairs is:

F(k j) Q

_ F(kj) l _ HY(co) N

1

(k - 2) .

Q

(A-10)

For the relevant conformations 2 f (k, j) = F,(k). I=1

Hence A, may be corrected by adding to it the quantity A; = ;

11 - l,7.JNig F,(k) -I

k=2

[

1 - Hq

(k - 2)]

N-g =

Q,/Q

-

[ ‘21

HY(~)

C k-2

Fe(k)

(k

-

2).

This expression is based on the assumption that the two helices containing the j and k - j base pairs remain optically independent. If it is supposed (as seems more likely) that the two helices joined by the stacking interaction behave optically as a single helix, the correction to A, takes the form:

-;

;i; E”,(k) [ 1 - Hq I =

(k - 2)]}.

(A-12)

The preceding discussion leads immedirttely to the contribution of conformations in which an interior loop divides the helical base pairs into two independent stacks:

HELIX

FORMATION

BY

1-y2+l) j7,(k)[I ._-!T! A3=-t! =-

OLIOOMERS.

1AD

II

(k - q]

Q3-~

N-(9/2+1)

HY(~)

F,(k)

Lc k=2

iI’&

Q Finally, the contribution

d(TA)

(k

-

2).

(A-13)

of the totally melted form (species I in Fig. 1) is (A-14)

A, = l/Q. The total relative absorbance is, then: A,,,(T)

= A, + A, +

(A-15)

+ A, + As.

A2

The analogy between this expression and equation (16) is enhanced by noting that

l/Q11+ Ql + Qz+ Qc+ Qal= 1. The relative proportions of conformations of various sorts as displayed in Figure 11 are simple ratios of partition functions. In particular.

&A/Q = &2/Q = QdQ = P/Q

J’, = (1 +

(A-15&)

f’,

(A-15b)

Ps

(A-15~)

(A-15d) P, where P is defined in equations (11) to (13). As shown in Figure 11 melting curves calculated for the simplest model of a single hairpin with a loop of constant minimum size are subject to negligible correction by inclusion of “higher order” conformations. It is interesting to note in Figure 12, however, that there is a substantial departure from the strict conditions of the

I

0

I

20

I

I

I

40

I

I

60

Temperature

I

80

I

I

100

(“C)

FIG. 11. The effect of including species IV to VI (Fig. 1) in predicting the melting curves for open hairpine. The solid linea show the melting curvea when only species I to III are oonsidered (aa in Fig. 6) while the dashed lines show the effects of allowing the loop eize to be variable and an additional loop to be present, in the manner illustr&ed in Fig. 1. (-) Species I to III; (- - - -) species I to VI.

170

I. E. SCHEFFLER,

E. L. ELSON

AND

R. L.

BALDWIN I

i ----r----i

Temperature (‘C) Fro. 12. For the linear oligomer

d(TA),,

the relative

proportions

are shown of species I to III

(P,) which have at most one loop of fixed size, of species I to IV (PI) with at most one loop which may be larger than the minimum-size loop, of species VI (P2) which has two loops of variable size, and of species V (Pa) which has an end loop and an interior loop, both of variable size. See Fig. 1.

simplest model. Although the simplest hairpins (P,) remain a majority at all temperatures, their number is substantially reduced in the transition region by the opening of interior and the enlargement of hairpin loops. This effect has little influence on the calculated melting curves because the loops must remain small. It may also be of interest to point out that the correction of Qa for the extra stacking interaction is important in the estimation of the number of double hairpins at low temperatures where essentially all of the few members of this species are stabilized by it. The correction has minor effect in the transition region and is of negligible importance to the melting curve at any temperature. In the calculations performed to study this question, a probable upper bound was used for rr (T*= 166: see page 167. We have profited from the discussion of this work with several people who may not, however, agree with everything we say here. In particular, we would like to acknowledge discueaione of the co-operativity of helix formation with Dr J. A. Scholhnan, of polynucleotide chain conformations with Drs P. J. Flory and G. Feleenfeld, of base stacking with Dr K. E. Van Holde, and also to thank Dr D. M. Crothera for his comments on an earlier versionof themanuscript. One of us (E. L. E.) gratefully acknowledges the encouragement and support (through National Institutes of Health training grant GM-01045) of Dr B. H. Zimm during much of this work. Another author (R. L. B.) wishes to acknowledge support from National Science Foundation and National Institutes of Health research grants (GB 4061 and AM 04763, respectively). REFERENCES Applequiet, p. 403. Applequist, Brahms, J.,

J. (1967). In Conformdone of Biopolymere, ed. by G. N. Ramaohandran, New York: Academic Press. J. t Damle, V. (1965). J. Amer. Chem. Sot. 87, 1450. Maurizot, J. C. & Michelson, A. M. (1967). J. Mol. Bid. 25, 481.

HELIX

FORMATION

BY d(TA)

OLlGOMERS.

II

171

Brandts, J. F. & Hunt, L. (1967). J. Amer. Chem. Sot. 89,4826. Crothere, D. & Zimm, B. H. (1964). J. MOE. Biol. 9, 1. Inetrumenta, Danforth, R., Krakauer, H. & Sturtevant, J. M. (1967). Review of Scknt@ as, 484. Davis, R. C. I% Tinoco, I., Jr. (1968). Bkpdymer8, 6, 223. DeVoe, H. BETinoco, I. (1962). J. Mol. BioZ. 4, 618. Eisenberg, H. & Felsenfeld, G. (1967). J. Mol. Bzbl. 30, 17. Elson, E. & Jovin, T. (1969). Andyt. Biochem. 27, 193. Felsenfeld, G. & Miles, H. T. (1967). Ann. Rev. Bhchem. 36, 407. Fisher, M. F. (1966). J. Chem. Phys. 45, 1469. Flory, P. J. (1963). PtincipZee of Polymer Chmhy. Cornell University Press. Flory, P. J. & Semlyen, J. A. (1966). J. Amer. Chem. Sot. 88, 3209. Imnan, R. B. t Baldwin, R. L. (1962). J. Mol. BioZ. 5, 172. Inman, R. B. & Baldwin, R. L. (1964). J. Mol. BioZ. 8, 452. Jacobson, H. & Stockmayer, W. H. (1950). J. Chem. Phye. 18, 1600. Kallenbach, N. (1968). J. Mol. BioZ. 87, 446. Kauzmann, W. (1959). Advunc. Protein Chem. 14, 1. Krakauer, H. t Sturtevant, J. M. (1968). RiopoZymers, 6, 491. Lehman, I. R. t Nussbaum, A. L. (1964). J. Bkl. Ch.em. 239, 2628. Magee, W. S., Gibbs, J. H. & Zimm, B. H. (1963). Biopolymere, 1, 133. Olivera, B. M. & Lehman, I. R. (1967). PTOC. Nat. Acd Sci,, Wash. 57, 1426. Olivera, B. M., Scheffler, I. E. & Lehman, I. R. (1968). J. Mol. Bid. 36, 291. Schachmann, H. K., Adler, J., Radding, C. M., Lehman, I. R. t Kornberg, A. (1060). J. BioZ. Chem. 235, 3242. Scheffler, I. E., Elson, E. L. & Baldwin, R. L. (1968). J. Mol. BioZ. 36, 291. Scheffler, I. E. & Sturtevant, J. M. (1969). J. Mol. BioZ. 42, 577. Tuppy, H. & Kiichler, E. (1964). Biochim. biophye. Acta, 80, 669. Wang, J. C. & Davidson, N. (1966). J. Mol. BioZ. 19, 469. Zimm, B. H. (1960). J. Chem. Phye. 33, 1349.